Normal modes for a stratified viscous fluid layer

2002 ◽  
Vol 132 (3) ◽  
pp. 611-625 ◽  
Author(s):  
K. F. Gurski ◽  
R. L. Pego
Keyword(s):  
Stokes Equations ◽  
Fluid Layer ◽  
Normal Modes ◽  
Internal Gravity Waves ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Damped Oscillations ◽  
Small Constant ◽  
Constant Viscosity ◽  
Stratified Viscous Fluid

We consider internal gravity waves in a stratified fluid layer with rigid horizontal boundaries and periodic boundary conditions on the sides at constant temperature with a small constant viscosity, modelled using the incompressible Navier-Stokes equations. Using operator-theoretic methods to study the damping rates of internal waves we prove there are non-oscillatory wave modes with arbitrarily small damping rates. We provide an asymptotic approximation for these non-oscillatory modes. Additionally, we find that the eigenvalues for damped oscillations are in an explicitly describable half-ring.

Internal laminar flow

2018 ◽  
Author(s):  
Marcel Escudier
Keyword(s):  
Poiseuille Flow ◽  
Stokes Equations ◽  
Circular Cross Section ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Concentric Cylinder ◽  
Navier Stokes Equations ◽  
Concentric Cylinders ◽  
Constant Viscosity ◽  
Pressure Driven Flow

In this chapter it is shown that solutions to the Navier-Stokes equations can be derived for steady, fully developed flow of a constant-viscosity Newtonian fluid through a cylindrical duct. Such a flow is known as a Poiseuille flow. For a pipe of circular cross section, the term Hagen-Poiseuille flow is used. Solutions are also derived for shear-driven flow within the annular space between two concentric cylinders or in the space between two parallel plates when there is relative tangential movement between the wetted surfaces, termed Couette flows. The concepts of wetted perimeter and hydraulic diameter are introduced. It is shown how the viscometer equations result from the concentric-cylinder solutions. The pressure-driven flow of generalised Newtonian fluids is also discussed.

scholarly journals Exploratory Simulation of Solar Granules: How Sharp is the Convection/Radiation Transition?

1998 ◽  
Vol 185 ◽  
pp. 217-218
Author(s):  
Kwing L. Chan ◽  
Y.C. Kim
Keyword(s):  
Stokes Equations ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Thermal Relaxation ◽  
3D Models ◽  
Numerical Approach ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Space Resolution ◽  
Radiation Transition

Currently, the most successful direct simulation of the solar granules (and the convection/radiation transition layer) is the three-dimensional (3D) model computed by Stein and Nordlund (1989). So far, there is no other similar 3D models available for comparison [however, see Ludwig et al. (1997) for a recent 2D calculation]. We are developing an alternative numerical approach to simulate the 3D radiation hydrodynamics of this layer. In this approach, the Eddington approximation is used to handle the radiation rather than solving the radiative transfer equations along rays, and the ADISM method (Chan and Wolff 1982) which solves the Navier Stokes equations in conservative forms is used to speed up the thermal relaxation of the fluid layer. We are in the process of testing the numerical accuracy of the codes. This paper summarizes the results of a test that illustrate the effects of vertical space resolution on the mean profiles of some important quantities.

Transient inertial hydrodynamic interaction between two identical spheres settling at small Reynolds number

2008 ◽  
Vol 605 ◽  
pp. 263-279 ◽  
Author(s):  
B. U. FELDERHOF
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Relative Motion ◽  
Hydrodynamic Interaction ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Small Reynolds Number ◽  
Navier Stokes Equations ◽  
Small Constant ◽  
Distance Vector

The flow pattern generated by a sphere accelerated from rest by a small constant applied forceshows scaling behaviour at long times, as can be shown from the solution of the linearized Navier–Stokes equations. In the scaling regime the kinetic energy of the flow grows with thesquare root of time. For two distant settling spheres starting from rest the kinetic energy ofthe flow depends on the distance vector between centres; owing to interference of the flowpatterns. It is argued that this leads to relative motion of the two spheres. Thecorresponding interaction energy is calculated explicitly in the scaling regime.

Film flow over heated wavy inclined surfaces

2010 ◽  
Vol 665 ◽  
pp. 418-456 ◽  
Author(s):  
S. J. D. D'ALESSIO ◽  
J. P. PASCAL ◽  
H. A. JASMINE ◽  
K. A. OGDEN
Keyword(s):  
Surface Tension ◽  
Stokes Equations ◽  
Fluid Layer ◽  
Bottom Topography ◽  
Marangoni Effect ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Inclined Surfaces ◽  
Variable Surface ◽  
The Stability

The two-dimensional problem of gravity-driven laminar flow of a thin layer of fluid down a heated wavy inclined surface is discussed. The coupled effect of bottom topography, variable surface tension and heating has been investigated both analytically and numerically. A stability analysis is conducted while nonlinear simulations are used to validate the stability predictions and also to study thermocapillary effects. The governing equations are based on the Navier–Stokes equations for a thin fluid layer with the cross-stream dependence eliminated by means of a weighted residual technique. Comparisons with experimental data and direct numerical simulations have been carried out and the agreement is good. New interesting results regarding the combined role of surface tension and sinusoidal topography on the stability of the flow are presented. The influence of heating and the Marangoni effect are also deduced.

Three-dimensional spatial normal modes in compressible boundary layers

2007 ◽  
Vol 586 ◽  
pp. 295-322 ◽  
Author(s):  
ANATOLI TUMIN
Keyword(s):  
Stokes Equations ◽  
A Priori ◽  
Three Dimensional ◽  
Normal Modes ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Compressible Boundary Layers ◽  
Partial Data ◽  
Finite Growth ◽  
The Cauchy Problem

Three-dimensional spatially growing perturbations in a two-dimensional compressible boundary layer are considered within the scope of linearized Navier–Stokes equations. The Cauchy problem is solved under the assumption of a finite growth rate of the disturbances. It is shown that the solution can be presented as an expansion into a biorthogonal eigenfunction system. The result can be used in a decomposition of flow fields derived from computational studies when pressure, temperature, and all the velocity components, together with some of their derivatives, are available. The method can also be used if partial data are available when a priori information may be utilized in the decomposition algorithm.

From onset of unsteadiness to chaos in a differentially heated square cavity

1998 ◽  
Vol 359 ◽  
pp. 81-107 ◽  
Author(s):  
PATRICK LE QUÉRÉ ◽  
MASUD BEHNIA
Keyword(s):  
Stokes Equations ◽  
Internal Gravity Waves ◽  
Chaotic Regime ◽  
Time Dependent ◽  
Navier Stokes ◽  
Transfer Coefficients ◽  
Navier Stokes Equations ◽  
Pressure Formulation ◽  
Order Of Magnitude ◽  
Spatial Approximation

We investigate with direct numerical simulations the onset of unsteadiness, the route to chaos and the dynamics of fully chaotic natural convection in an upright square air-filled differentially heated cavity with adiabatic top and bottom walls. The numerical algorithm integrates the Boussinesq-type Navier–Stokes equations in velocity–pressure formulation with a Chebyshev spatial approximation and a finite-difference second-order time-marching scheme. Simulations are performed for Rayleigh numbers up to 1010, which is more than one order of magnitude higher than the onset of unsteadiness. The dynamics of the time-dependent solutions, their time-averaged structure and preliminary results concerning their statistics are presented. In particular, the internal gravity waves are shown to play an important role in the time-dependent dynamics of the solutions, both at the onset of unsteadiness and in the fully chaotic regime. The influence of unsteadiness on the local and global heat transfer coefficients is also examined.

scholarly journals Regularity of Weak Solutions to the Inhomogeneous Stationary Navier–Stokes Equations

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1336
Author(s):  
Alfonsina Tartaglione
Keyword(s):  
Bounded Domain ◽  
Weak Solutions ◽  
Analogous Result ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Hölder Continuous ◽  
Navier Stokes Equations ◽  
Regularity Of Weak Solutions ◽  
Constant Viscosity ◽  
Stationary Navier Stokes Equations

One of the most intriguing issues in the mathematical theory of the stationary Navier–Stokes equations is the regularity of weak solutions. This problem has been deeply investigated for homogeneous fluids. In this paper, the regularity of the solutions in the case of not constant viscosity is analyzed. Precisely, it is proved that for a bounded domain Ω⊂R2, a weak solution u∈W1,q(Ω) is locally Hölder continuous if q=2, and Hölder continuous around x, if q∈(1,2) and |μ(x)−μ0| is suitably small, with μ0 positive constant; an analogous result holds true for a bounded domain Ω⊂Rn(n>2) and weak solutions in W1,n/2(Ω).

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